Midterm Exam 1, section 1. Tuesday, February hour, 30 minutes

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1 San Francisco State University Michael Bar ECON 312 Spring 2018 Midterm Exam 1, section 1 Tuesday, February 20 1 hour, 30 minutes Name: Instructions 1. This is closed book, closed notes exam. 2. You can use one double-sided sheet of paper, letter size (8½ 11 in or mm), with any content you want. 3. No calculators of any kind are allowed. 4. Show all the calculations, and explain your steps. 5. If you need more space, use the back of the page. 6. Fully label all graphs. Good Luck

2 1. (10 points). Let XX be a random variable with mean μμ and variance 2, and let XX μμ YY = be the standardized transformation of XX. a. Using the rules of expected values show that the mean of YY is 0. XX μμ EE(YY) = EE given form of YY = 1 EE(XX μμ) constants factor out of EE = 1 (EE(XX) μμ) EE of sum = sum of EE = 1 (μμ μμ) it is given that EE(XX) = μμ = 0 b. Using the rules of variances, show that the variance of YY is 1. XX μμ vvvvvv(yy) = vvvvvv given the form of YY = 1 vvvvvv(xx μμ) constants factor out of vvvvvv squared 2 = 1 vvvvvv(xx) adding constant does not affect vvvvvv 2 = = 1 given vvvvvv(xx) 1

3 2. (10 points). Let XX and YY be two random variables. Prove that cccccccc(2xx, 3YY) = cccccccc(xx, YY) Using the definition of correlation, and rules of variance and covariance, we get the following steps: cccccc(2xx, 3YY) cccccccc(2xx, 3YY) = vvvvvv(2xx) vvvvvv(3yy) 2 3 cccccc(xx, YY) = 4vvvvvv(XX) 9vvvvvv(YY) 2 3 cccccc(xx, YY) = = cccccccc(xx, YY) 2 3 vvvvvv(xx) vvvvvv(yy) 2

4 3. (20 points). Let XX 1, XX 2,, XX be a random sample from population XX, with population mean μμ and variance 2. a. Prove that (XX ii XX ) = 0 where XX is the sample average. Your answer must start with a definition of sample average. Sample average is defined as follows: Thus, XX = 1 XX ii (XX ii XX ) = XX ii XX = XX XX = 0 b. Supposed that EE(XX 1 ) = 7 and vvvvvv(xx 1 ) = 4. Find EE(XX 7 ) and vvvvvv(xx 5 ). Since XX 1, XX 2,, XX is a random sample, all observations must have the same distribution. Thus, EE(XX ii ) = 7 ii and vvvvvv(xx ii ) = 4 ii. In particular, EE(XX 7 ) = 7 and vvvvvv(xx 5 ) = 4. 3

5 4. (20 points). In order to estimate the population mean, a random sample of observations was collected XX 1, XX 2,, XX, and the sample average XX = 1 XX ii is proposed as an estimator. a. Prove that XX is an unbiased estimator of the population mean μμ. EE(XX ) = EE 1 XX ii = 1 EE(XX ii) = 1 μμ = 1 = μμ b. Let 2 denote the population variance. Prove that XX is a consistent estimator of the population mean μμ. Since we proved that XX is unbiased, we only need to prove that lim vvvvvv(xx ) = lim 2 = 0. vvvvvv(xx ) = vvvvvv 1 XX ii = 1 2 vvvvvv(xx ii) = = 2 lim vvvvvv(xx ) = lim 2 = 0 4

6 5. (20 points). Consider the simple regression model YY ii = ββ 1 + ββ 2 XX ii + uu ii. a. Suppose that YY ii is GPA (Grade Point Average) of student ii, and XX ii is attendance, measured as the number of classes skipped per week. What is the interpretation of the error term uu ii? The error term uu ii represents all the factors, other than attendance, which affect student s GPA. For example, the student s major, working habits, ability, work environment, quality of high school education, etc. b. Define the OLS estimators of the unknown parameters ββ 1, ββ 2 and denote them by bb 1 OOOOOO, bb 2 OOOOOO. Let the fitted model be YY ii = bb 1 + bb 2 XX ii, where bb 1 and bb 2 are some estimates of ββ 1 and ββ 2. The residual of observation ii (or prediction error) is ee ii = YY ii YY ii. The OLS estimators bb OOOOOO OOOOOO 1, bb 2 are values of bb 1, bb 2 which minimize the Residual Sum of Squares, i.e. solve the following problem: 2 min RRRRRR = ee ii = (YY bb 1,bb 2 ii bb 1 bb 2 XX ii ) 2 5

7 c. Suppose that Ariana estimated bb 1 = 3 and bb 2 = 0.2. What is the predicted GPA of a student who skips 2 classes per week? Substituting the given values into the fitted equation: YY ii = bb 1 + bb 2 XX ii = = = 2.6 d. If the average GPA in Ariana s sample is 2.7, what is the average number of classes skipped in the sample? Using the fact that the fitted equation must pass through the point of sample averages, YY = bb 1 + bb 2 XX 2.7 = 3 0.2XX 0.2XX = 0.3 XX = 1.5 classes per week 6

8 6. (20 points). Jessica is studying the relationship between fertility and contraception prevalence in developing countries. She collected data on 50 countries/regions The key variables are: tfr total fertility rate (number of children per woman) contraceptors - Percent (%) of married women of childbearing age, who use contraception (i.e. devices that prevent pregnancy) Jessica s R output is presented below: lm(tfr ~ contraceptors, data = contraception) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** contraceptors <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 48 degrees of freedom Multiple R-squared: 0.847, Adjusted R-squared: F-statistic: on 1 and 48 DF, p-value: < 2.2e-16 a. What is the dependent variable in the above regression model? tfr b. What is the independent variable (regressor) in the above regression model? contraceptors 7

9 c. Interpret the estimated regression coefficients. bb 2 = means that a 1% increase in the percent of women using contraception, lowers total fertility rate in that country (or region) by children per woman. bb 1 = is the predicted number of children in a country (or region) with no contraception use. d. Explain the meaning of the reported RR 2, and comment on its magnitude. RR 2 = means that nearly 85% of the variation in total fertility rate in the sample can be explained by the model, with contraceptors as the only regressor. This magnitude is very high, and it means that all other influences on fertility, such as culture, women s education, level of industrialization, all together account for only 15% of the variation in fertility. Remark: No doubt that contraceptives, if used, are effective to prevent pregnancy. However, in some countries contraceptives are not widely used. The use of contraception in a country may represent other factors, such as culture, women s education, etc. 8